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The aim of this course is to give an introduction to the statistical models on random planar maps which can be mapped to ensembles of loops and lines on the planar map, such as the O(n) loop model as well as the models enjoying sl(2) quantum group symmetry: the height RSOS and A-D-E models and the 6-vertex model. The loop models on planar maps can be studied by analytic combinatorics or by being reformuated as large-N matrix models. The solution is formulated in terms of a spectral curve, in general non-algebraic. The singular points of the spectral curve describe universal scaling limits of random planar map models. The nature of the singularities reflects the properties of large weighted maps and can be understood in the framework of Liouville quantum gravity. In this sense the loop models on random planar maps represent discretisation of two- dimensional quantum gravity with continuous spectrum of the Virasoro central charge for the matter field.

1. The O(n) loop model on random maps. Maps and loop configurations. Statistical weights and partition functions. Phase diagram and critical points. Combinatorial solution: nested loop approach, loop equations, transfer matrix, functional equations. O(n) matrix model. Spectral curve, critical behaviour and relation to Liouville gravity. 2. SOS, RSOS and A-D-E models on planar graphs. Statistical weights and partition functions. Formulation in terms of loop gas. Combinatorial solution by nested loop approach. Spectral curve, critical behaviour and relation to Liouville gravity. The A-D-E matrix models as minimal models of 2D quantum gravity. The SOS model on random maps and loop ensembles associated with affine Lie algebras of A-D-E type. 3. The six-vertex model on random maps. Statistical weights and partition functions. Reformulation as a loop model and combinatorial solution. The six-vertex matrix model. Critical behaviour and relation to Matrix Quantum Mechanics.

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